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User blog:Eners49/New Notation Idea? (Attempt 2)
Here is a new notation I am thinking about. I think i already reached w^w in FGH, which was my initial goal; now my goal is to hit epsilon-zero. When I saw PlantStar's Notation and how easy it was to create your own notation, I got the inspiration to create my own. However, the first time I tried, I utterly and completely fucking failed. I extended upon Graham's function, which has growth w+1, but I did a bunch of complicated stuff and only got to w+9 :( For my second attempt, I will try to make my own version of BEAF, based off of the factorial function. Right now, I have not developed very much and my notation is not very strong yet, but it will be! I promise. Here is the basics: *{a} = a!!!...!!! with a factorial signs. Pretty basic for 1-entry arrays. Similar to Aarex's warp notation. *{0} = 0 *{1} = 1! = 1 *{2} = 2!! = 2! = 2 *{3} = 3!!! = 6!! = 720! = 10^1,746.42 *{4} = 4!!!! = 24!!! = 10^10^10^24 *In general, {a} is about 10^^a With one-entry arrrays, we already have created a strong function, but we're only in between \(f_2(n)\) and \(f_3(n)\) in the fast-growing hierarchy, so we need to keep going. Let's see what two-entry arrays look like: *{a, b} = }...}}}, b-1} where there are a brackets. We've created a pretty strong recursion so far. *{a, 1} = {a}. Just like BEAF, if there are any 1's at the end, they can be removed. *{1, 2} = }, 1} = }} = } = { {10^^10^^10^^10^^(10^1,750)}} *{4, 2} = }}, 1} = }}} = }} = 10^^10^^10^^10^^(10^10^10^24) *In general, {a, 2} is about 10^^^a *{3, 3} = }, 2} = {10^^10^^10^^(10^1,750), 2} = }, 1} = about 10^^^10^^^(10^1,750) *In general, {a, 3} is about 10^^^^a *In general, {a, b} = 10^^^...^^^a with b+1 arrows Wow! With just two entries, we have created something that grows as fast as Knuth's arrows do. Thus, we have hit growth rate w in the fast-growing hierarchy, starting from scratch! This is as powerful as the Ackermann function and three-entry BEAF. But the three-entry arrays in my notation are way crazier. *{a, b, c} = }...}}}, b-1, c} with a bracket sets *{a, 1, c} = {a, a, c-1} So {a, b, 2} has growth rate w*2, {a, b, 3} has growth rate w*3, etc. so the limit of 3-entry arrays is w^2. Leaving my previous record of w+9 IN THE DUST! With 4+ entries, it's very similar to normal BEAF (thanks for the suggestion Syst3ms): # signifies the rest of the array. *{a,1,...,1,c+1,#} = {a,1,...,a,c,#} So 4-entry arrays have growth rate w^3, 5-entry arrays have growth rate w^4, etc., and a-entry arrays have growth rate w^(a-1). So we can say that the limit of this is w^w. But beyond this, it's a little different. *{a | b} = {a, a, a, ..., a, a} with b a's *{a, 2 | b} = the following: *Stage 1 = {a | b} = {a, a, a, ..., a, a} *Stage 2 = {{a | b} | b} = {a, a, a, ..., a, a} with Stage 1 a's *Stage 3 = {{{a | b} | b} | b} = {a, a, a, ..., a, a} with Stage 2 a's *... *{a, 2 | b} = Stage a Insane, right?? *{a, 3 | b} = the following: *Stage 1 = {a, 2 | b} *Stage 2 = {{a, 2 | b}, 2 | b} *Stage 3 = {{{a, 2 | b}, 2 | b}, 2 | b} *... *{a, 3 | b} = Stage a {a, 4 | b}, {a, 5 | b}, etc. should be obvious. Well, that's it for now. I promise, I will extend this further in the future! Please, anyone who has experience with fast-growing hierarchies and stuff, tell me the maximum growth rate of my notation. I suck at fast-growing hierarchies and probably a lot of what I have written in here is complete and utter nonsense. Someone pls help me Category:Blog posts